The continuous Zak transform is a mapping of a continuous-time signal onto a two-dimensional function. Implicit usage of the Zak transform can be traced back until Gauss [1]}; however, it was Zak who formally introduced the transform in [2]} and after whom it was named. An excellent paper from a signal theoretical point of view was provided by Janssen [3]}. Later on, Bölcskei and Hlawatsch [4]} provided an overview of the discrete versions of the transform: the discrete-time Zak transform and the discrete Zak transform. This section is devoted to the DZT and its properties since we will use it to describe OTFS and to establish the input-output relation of TF dispersive channel in Sec. .
